Chapter 5: Problem 105
When \(1.034 \mathrm{~g}\) of naphthalene \(\left(\mathrm{C}_{10} \mathrm{H}_{8}\right)\) is burned in a constant-volume bomb calorimeter at \(298 \mathrm{~K}, 41.56 \mathrm{~kJ}\) of heat is evolved. Calculate \(\Delta U\) and \(w\) for the reaction on a molar basis.
Short Answer
Expert verified
\( \Delta U = 5150 \text{ kJ/mol} \) and \( w = 0 \).
Step by step solution
01
Identify Given Data
We are given that 1.034 g of naphthalene (C₁₀H₈) is burned and releases 41.56 kJ of heat. The reaction occurs at 298 K.
02
Calculate Molar Mass of Naphthalene
Naphthalene (C₁₀H₈) consists of 10 carbon atoms and 8 hydrogen atoms. The molar mass is calculated as follows: \[ \text{Molar mass of C} = 12.01 \text{ g/mol}, \text{Molar mass of H} = 1.008 \text{ g/mol} \] Therefore, \[ \text{Molar mass of naphthalene} = (10 \times 12.01) + (8 \times 1.008) = 128.18 \text{ g/mol} \]
03
Calculate Moles of Naphthalene
Use the mass of naphthalene to calculate moles: \[ \text{Moles of naphthalene} = \frac{1.034 \text{ g}}{128.18 \text{ g/mol}} = 0.00807 \text{ mol} \]
04
Calculate Internal Energy Change (ΔU)
Since the bomb calorimeter's experiment is at constant volume, \( \Delta U \) is equivalent to the heat evolved. Therefore, \( \Delta U \) per mole can be calculated by dividing the total heat transferred by moles: \[ \Delta U = \frac{41.56 \text{ kJ}}{0.00807 \text{ mol}} = 5150 \text{ kJ/mol} \]
05
Calculate Work (w) for the Reaction
In a constant volume scenario, no work is done as work (w) is defined as \( w = -P\Delta V \). Since \( \Delta V = 0 \) at constant volume, \( w = 0 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Internal Energy
Internal energy is the total energy contained within a system. In the realm of thermodynamics, it includes both the kinetic energy from the movement of particles and the potential energy from forces acting between them. For chemical reactions, the internal energy, denoted as \( \Delta U \), is especially important as it reflects how energy changes during the reaction.
In a constant volume process, such as in a bomb calorimeter, there is no work done by the system. Thus, the change in internal energy, \( \Delta U \), directly equates to the heat absorbed or released. For example, when naphthalene is burned, it releases 41.56 kJ of energy, which is accounted for in the change of its internal energy. The equation \( \Delta U = q_{v} \) is used in such cases, where \( q_{v} \) is the heat at constant volume.
In a constant volume process, such as in a bomb calorimeter, there is no work done by the system. Thus, the change in internal energy, \( \Delta U \), directly equates to the heat absorbed or released. For example, when naphthalene is burned, it releases 41.56 kJ of energy, which is accounted for in the change of its internal energy. The equation \( \Delta U = q_{v} \) is used in such cases, where \( q_{v} \) is the heat at constant volume.
Bomb Calorimeter
A bomb calorimeter is a device used to measure the amount of heat absorbed or released during a chemical reaction at constant volume. It encapsulates the reaction in an insulated environment, ensuring no heat is lost to the surroundings.
This setup is particularly useful for reactions such as combustion, where gases and heat are involved. During an experiment with a bomb calorimeter, the substance is placed in a small container, called a bomb, and is then surrounded by water or another heat-absorbing medium. The calorimeter records changes in thermal energy, allowing calculations of the internal energy change, \( \Delta U \).
A bomb calorimeter provides critical insights into energy dynamics as no work is done (\( w = 0 \)) at constant volume.
This setup is particularly useful for reactions such as combustion, where gases and heat are involved. During an experiment with a bomb calorimeter, the substance is placed in a small container, called a bomb, and is then surrounded by water or another heat-absorbing medium. The calorimeter records changes in thermal energy, allowing calculations of the internal energy change, \( \Delta U \).
A bomb calorimeter provides critical insights into energy dynamics as no work is done (\( w = 0 \)) at constant volume.
Molar Mass
Molar mass is a fundamental concept in chemistry, representing the mass of one mole of a substance, usually expressed in grams per mole (g/mol). It is calculated by summing up the atomic masses of all atoms present in a molecule.
For instance, with naphthalene, \( \text{C}_{10} \text{H}_{8} \), the molar mass is determined by adding the molar masses of the individual carbon and hydrogen atoms. Given that the molar mass of carbon is 12.01 g/mol and hydrogen is 1.008 g/mol, naphthalene has a molar mass of 128.18 g/mol. Calculating molar mass allows scientists to convert grams of a substance into moles, which is a crucial step in stoichiometric calculations.
For instance, with naphthalene, \( \text{C}_{10} \text{H}_{8} \), the molar mass is determined by adding the molar masses of the individual carbon and hydrogen atoms. Given that the molar mass of carbon is 12.01 g/mol and hydrogen is 1.008 g/mol, naphthalene has a molar mass of 128.18 g/mol. Calculating molar mass allows scientists to convert grams of a substance into moles, which is a crucial step in stoichiometric calculations.
Chemical Reaction
Chemical reactions are processes where reactants transform into products through the breaking and formation of chemical bonds. In the context of thermodynamics, chemical reactions involve energy changes, manifested as either heat absorption or release.
When naphthalene undergoes combustion in a bomb calorimeter, it is a reaction where hydrocarbons react with oxygen to produce carbon dioxide, water, and heat. This reaction is exothermic, releasing energy.
When naphthalene undergoes combustion in a bomb calorimeter, it is a reaction where hydrocarbons react with oxygen to produce carbon dioxide, water, and heat. This reaction is exothermic, releasing energy.
- Exothermic reactions: Reactions that release heat, such as combustion.
- Endothermic reactions: Reactions that absorb heat, requiring an input of energy.